Lemma 3.2 (Elekes and Rónyai [1]}) 
Let \(U\subset \mathbb {R}^2\)  be a connected open set and let \(P\colon U\rightarrow \mathbb {R}\)  be analytic. If \(P_x\)  or \(P_y\)  vanishes identically on \(U\) , then \(P\)  is an analytic special form, in the sense of Definition REF . If neither \(P_x\)  nor \(P_y\)  vanishes identically on \(U\) , then \(P\)  is an analytic special form if and only if \(\partial _{xy}\left(\log \left(\frac{\partial _xP}{\partial _yP}\right)\right)\)  vanishes everywhere on \(U\)  that it is defined (i.e. everywhere \(\partial _xP\)  and \(\partial _yP\)  are non-zero).

